^{1,a)}, A. B. Rosenfeld

^{1}, R. W. Schulte

^{2}and K. E. Schubert

^{3}

### Abstract

An accurate system matrix is required for quantitative protonCT (pCT) image reconstruction with iterative projection algorithms. The system matrix is composed of chord lengths of individual proton path intersections with reconstruction pixels. In previous work, reconstructions were performed assuming constant intersection chord lengths, which led to systematic errors of the reconstructedprotonstopping powers. The purpose of the present work was to introduce a computationally efficient variable intersection chord length in order to improve the accuracy of the system matrix. An analytical expression that takes into account the discrete stepping nature of the pCT most likely path (MLP) reconstruction procedure was created to describe an angle-dependent effective mean chord length function. A pCT dataset was simulated with GEANT4 using a parallel beam of 200 MeV protons intersecting a computerized head phantom consisting of tissue-equivalent materials with known relative stopping power. The phantom stopping powers were reconstructed with the constant chord length, exact chord length, and effective mean chord length approaches, in combination with the algebraic reconstruction technique. Relative stopping power errors were calculated for each anatomical phantom region and compared for the various methods. It was found that the error of approximately 10% in the mean reconstructedstopping power value for a given anatomical region, resulting from a system matrix with a constant chord length, could be reduced to less than 0.5% with either the effective mean chord length or exact chord length approaches. Reconstructions with the effective mean chord length were found to be approximately 20% faster than reconstructions with an exact chord length. The effective mean chord length method provides the possibility for more accurate, computationally efficient quantitative pCT reconstructions.

This work was supported by Grant No. 08/RSA/1-02 from the Cancer Institute NSW.

I. INTRODUCTION

II. METHODS

II.A. Most likely path formalism

II.B. Exact chord length approach

II.C. Effective mean chord length approach

II.D. ProtonCT simulation

II.E. ProtonCTimage reconstruction and evaluation

III. RESULTS

III.A. Quantitative accuracy of protonCTreconstructions

III.B. Reconstruction time

IV. DISCUSSION

V. CONCLUSION

### Key Topics

- Protons
- 58.0
- Collisional energy loss
- 30.0
- Medical imaging
- 26.0
- Medical image reconstruction
- 24.0
- Image reconstruction
- 22.0

## Figures

Conceptual illustration of the MLP formalism. The bold line represents the MLP, while the faint line corresponds to a proton undergoing exaggerated multiple Coulomb scattering.

Conceptual illustration of the MLP formalism. The bold line represents the MLP, while the faint line corresponds to a proton undergoing exaggerated multiple Coulomb scattering.

Plot of the derived effective mean chord length as a function of pixel rotation angle.

Plot of the derived effective mean chord length as a function of pixel rotation angle.

Schematic of the GEANT4 simulation geometry used to model an ideal pCT system.

Schematic of the GEANT4 simulation geometry used to model an ideal pCT system.

(a) The Herman head phantom. Reconstructed images corresponding to the cycle of minimum relative error with (b) constant chord length, (c) exact chord length, and (d) effective mean chord length.

(a) The Herman head phantom. Reconstructed images corresponding to the cycle of minimum relative error with (b) constant chord length, (c) exact chord length, and (d) effective mean chord length.

Distribution of reconstructed relative stopping powers with constant chord length, exact chord length, and effective mean chord lengths.

Distribution of reconstructed relative stopping powers with constant chord length, exact chord length, and effective mean chord lengths.

Schematic of the rotated pixel geometry. The pixel vertices are denoted by points A, B, C, and D. The linear functions joining these points are labeled , , , and . A simplified straight line proton path is given as an example, illustrating the discrete stepping nature of the MLP. The step size is denoted by . The chord length for this example is shown in bold.

Schematic of the rotated pixel geometry. The pixel vertices are denoted by points A, B, C, and D. The linear functions joining these points are labeled , , , and . A simplified straight line proton path is given as an example, illustrating the discrete stepping nature of the MLP. The step size is denoted by . The chord length for this example is shown in bold.

Derivation of the point on the axis at which the chord length through a pixel is equal to the step size of the MLP procedure. Through symmetry, the same method can be used to derive for the positive axis.

Derivation of the point on the axis at which the chord length through a pixel is equal to the step size of the MLP procedure. Through symmetry, the same method can be used to derive for the positive axis.

## Tables

Results of a Guassian fit to histograms of reconstructed bone and brain regions in the relative stopping power images. The mean value with 95% confident limit and standard deviation are given.

Results of a Guassian fit to histograms of reconstructed bone and brain regions in the relative stopping power images. The mean value with 95% confident limit and standard deviation are given.

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